| | |
Summary: INTEGRAL PINCHING THEOREMS
XIANZHE DAI, PETER PETERSEN, AND GUOFANG WEI
Abstract. Using Hamilton's Ricci
ow we shall prove several pinching results
for integral curvature. In particular, we show that if p > n=2 and the L p norm
of the curvature tensor is small and the diameter is bounded, then the manifold
is an infra-nilmanifold. We also obtain a result on deforming metrics to positive
sectional curvature.
1. Introduction
The goal of this note is to prove several pinching results for manifolds with
integral curvature bounds. Integral pinching has been studied extensively in [3],
[2], [12], [13], [15], [8]. One distinct feature in our work is that assumptions on
curvature are entirely in terms of integral bounds and no assumption on volume,
injectivity radius or Sobolev constant is made.
Let us x some notation before we state the results. For a Riemannian manifold
(M; g), we will denote by sec : M ! R the minimum of the sectional curvature
at each point, by R : 2 TM ! 2 TM the curvature operator, by Ric the Ricci
curvature, and by Scal the scalar curvature.
For functions and tensors we shall consistently use the normalized L p norm
de ned by
kuk p =
|
